Canards interference on the Magnus effect of a fin-stabilized spinning missile

Abstract

Reynolds-averaged simulations of flow over spinning finned missiles with and without canards were carried out at Ma = 0.6, 0.9, 1.5, and 2.5; α = 4°, 8°, and 12.6°; and $\overline{\omega }=0.025$ to investigate different mechanisms of the Magnus effect. An implicit dual-time stepping method and the $\gamma -R{e}_{\theta t}$ transition model were combined to solve the unsteady Reynolds-averaged Navier–Stokes equations. Grid independence study was conducted, and the computed results were compared with archival experimental data. The transient and time-averaged lateral force coefficients were obtained, and the flow field structures were compared at typical rolling angles. The results indicate that in subsonic conditions, the canards interference intensifies the asymmetrical distortion of the body surface boundary layer and flow separation at different angles of attack, doubling the absolute value of the time-averaged body lateral force; the wash flow effect strengthens on the leeward tail due to the canards interference, increasing its time-averaged lateral force; in supersonic conditions, the shock and expansion waves induced by canards, the vortex system, and the flow separation are responsible for the fluctuation of the body lateral force; the direction of the canard induced wash flow alters as angle of attack increases, increasing first and then decreasing the time-averaged tail lateral force.

Introduction

With the development of technology, modern war is becoming high-speed and flexible. The key points lie in increasing shooting range, improving accuracy, and enhancing damage effect on the premise of maintaining cost-effective ratio. After the Second World War, the development of artillery rocket weapons experienced two stages. One was the free-flight stage aiming at increasing shooting range and the other was the simple controlled stage to decrease shooting dispersion. Up to now, it has come to a guided stage to enhance shooting accuracy. The external configuration of artillery rockets usually consists of canards, cylindrical body with large slenderness ratio, and axial symmetrical tails. Spin is adopted to eliminate the adverse effects of mass, thrust, and aerodynamic eccentricity. Meanwhile, the control system is simplified, and the pitching and yawing motions can be realized through a pair of canards.

When the axial symmetrical vehicle is flying around its longitudinal axis at angles of attack, the combination of crossflow and spin can induce additional lateral force and yawing moment, that is, the Magnus effect. ¹ Many researches on the Magnus effect of non-finned and finned projectiles and missiles have been carried out. The dominating flow mechanism of the Magnus effect of non-finned projectiles includes the surface boundary layer distortion, the asymmetrical flow separation, and the asymmetrical transition. ² For finned projectiles, additional angle of attack of tails induced by spin rate and the impact of fore-body separation vortices on tails would produce additional influence.^3,4 Although the lateral force is only 1/100 to 1/10 of the normal force, the coupled longitudinal and lateral motions can be induced and is characterized as coning motion. Coning motion can increase drag and decrease shooting range. When lateral force and yawing moment exceed certain limit, the divergent coning motion may directly lead to flight failure. ⁵

An important advantage of canards is to improve control efficiency. Many researches have been carried out to investigate the aerodynamic characteristics and the flight stability of spinning finned projectiles and missiles with canards. The aerodynamic characteristics of a 155-mm guided canard projectile were obtained by Su et al. ⁶ through computational fluid dynamics (CFD) computations and wind tunnel tests. Silton and Fresconi ⁷ investigated the interference of the canards on the aerodynamic characteristics and the flow field structures of a projectile in non-spin conditions. Zhu et al. ⁸ made study on the flight stability of the dual spin 155 artillery projectile and revised the stability criterion. The following studies were focused on configuration with large slenderness ratio. Wind tunnel tests were made by Eastman ⁹ to discuss the influence of canard existence, canard size, and canard rolling position on the aerodynamic characteristics of tail in non-spin conditions. The aerodynamic characteristics of a spinning finned missile with dithering canards were computed by Blades and Marcum ¹⁰ using unstructured overset grids at Ma = 1.6 and $\alpha =3°$ . The deflection angle of canard is within the range of $\delta \in (-15°,+15°)$ . The results indicated that the canard-induced separation vortex had profound influence on tail at certain rolling angle. Nygaard and Meakin ¹¹ followed the study of Blades and Marcum, while structured overset grids were used and the angle of attack was expanded to $\alpha =15°$ . The results showed that viscous effect was important in capturing the interference between the boundary layer and the separation vortex induced by canard. Sheng et al. ¹² calculated the aerodynamic characteristics of the above missile with fixed canards using unstructured overset grids. The computational conditions were Ma = 0.8–2.2 and $\alpha =0°-15°$ . In supersonic conditions, for canard with zero deflection angle, the vortex of canard tip left the body rapidly as angle of attack increased. However, canard root might continuously influence the flow around the missile body, the leeward separation vortex, and the forces acted on tail.

Problems were also brought when canards were introduced to improve the maneuverability of spinning vehicles, because canards can significantly interfere in the aerodynamic characteristics of the body and the tails. In the literature,^10–12 significant time-averaged lateral force and yawing moment were obtained when a pitching control was conducted. The sources of the time-averaged lateral force consist of the configuration asymmetry about angle of attack plane caused by the canards deflection and the Magnus effect induced by spin. The above references were focused on the computation of the whole aerodynamic characteristics, while the physical origins for the aerodynamic interference were not discussed in detail. Generally, the variation tendency and the flow mechanism of the lateral force and yawing moment of spinning finned missiles with canards are quite different from those of configurations without canards. So far, to the authors’ knowledge, few attentions have been paid on the numerical investigations of the Magnus effect of spinning finned missiles with canards.

In this study, two canards with zero deflection angle were added to the Air Force Modified Basic Finner Missile (AFF), ¹³ and the aerodynamic characteristics of the two configurations were computed at Ma = 0.6, 0.9, 1.5, 2.5; $\alpha =4°,8°,\mathrm{and}12°$ ; and $\overline{\omega }=0.025$ . The aerodynamic coefficients of the spinning missiles with and without canards were first and directly compared. The key lies in the reveal of the interference of canards on the body and tail lateral force and yawing moment through the comparison between flow field structures. For this symmetrical configuration, the time-averaged lateral force is totally attributed to the Magnus effect. The flow mechanism for the Magnus effect can be deduced through transient state analysis at different rolling angles. A profound insight into the flow structure and mechanism can provide guidance to the design of vehicle configuration and control system.

Numerical method

Governing equations and turbulence models

An unsteady Reynolds-averaged Navier–Stokes (URANS) code based on finite volume method was used for CFD computation. The integral form of the unsteady Navier–Stokes (N-S) equations were chosen as governing equations for fluid flow

$$\frac{\partial }{\partial t}\underset{V}{\overset{}{\int }}\mathbf{W}\mathrm{dV}+\mathrm{\Gamma }\frac{\partial }{\partial \epsilon }\underset{V}{\overset{}{\int }}\mathbf{Q}\mathrm{dV}+\oint [\mathbf{F}-\mathbf{G}]·d\mathbf{A}=\underset{V}{\int }\mathbf{H}\mathrm{dV}$$

(1)

$$\begin{array}{l}W=\left\{\begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ \rho E\end{array}\right\},Q=\left\{\begin{array}{c}p\\ u\\ v\\ w\\ T\end{array}\right\},F=\left\{\begin{array}{c}\rho v\\ \rho vu+pi\\ \rho vv+pj\\ \rho vw+pk\\ \rho vE+pv\end{array}\right\},\\ G=\left\{\begin{array}{c}0\\ {\tau }_{xi}\\ {\tau }_{yi}\\ {\tau }_{zi}\\ {\tau }_{ij}{v}_j+f\end{array}\right\}\end{array}$$

(2)

where H, F, and G are the source term, the convective term, and the viscous term, respectively. The N-S equations were decomposed by Reynolds averaging. The dual-time stepping method, consisting of a physical time step to describe the motion of the vehicle and an inner iteration time step to converge the RANS equations, was used for time-accurate calculations.^14,15 Second-order upwind scheme was used for spatial discretization. Roe flux-difference splitting scheme was used for convective fluxes. A steady-state calculation with zero spin rate was conducted first, which was deemed converged when the residual had dropped at least three orders of magnitude and the aerodynamic coefficients had varied within 0.1% during the last 500 iterations. The steady-state result was treated as an initial condition for unsteady calculation to rapidly get converged result. During the unsteady calculation, the inner iteration step was set to 20. ¹⁶ To ensure the reliability of the numerical results, grid independence study was carried out, and the results from two turbulence models were compared, including the $k-\omega$ turbulence model and the $\gamma -R{e}_{\theta t}$ transition model.^17–19

Computational models and grids

The AFF configuration consists of an ogive nose, a cylindrical body, and four symmetrical uncanted fins. ¹³ The canard configuration was obtained by adding two undeflected fixed canards on the front of the cylindrical body. The computational model and the relative axial position are shown in Figure 1. The diameter of the missile body is d = 0.04572 m, and the slenderness ratio is L/d = 10. The same computational conditions were used for the two configurations: $R{e}_d=2.57\times {10}^5$ ; Ma = 0.6, 0.9, 1.5, and 2.5; $\alpha =4°,8°,\mathrm{and}12.6°$ ; and $\overline{\omega }=0.025$ . The reference length and reference area are $L_{\mathrm{ref}}=0.04572\mathrm{m}$ and $S_{\mathrm{ref}}=0.00164{\mathrm{m}}^2$ , respectively. The distance between the moment reference point and the nose vertex is H = 0.2286 m.

Figure 1.

Model configuration and dimension.

The three-dimensional (3D) structured hexahedral grids are shown in Figure 2. The height of the first layer grid was $5\times {10}^{-6}\mathrm{m}$ to keep y+ ≤ 1. The grid inner boundary was set as no-slip adiabatic wall. For supersonic conditions, the grid outer boundary extends one projectile length in all directions and was set to freestream (nose and circumferential) and pressure outlet (base) conditions. For subsonic conditions, the grid outer boundary extends about 10 projectile lengths and was set to freestream conditions. To exclude the influence of grid on computational results, for each configuration, two meshes with different amount were chosen for grid independence study. The grid parameters around the missile surface for supersonic calculations are shown in Table 1.

Figure 2.

Computational grids: (a) AFF model and (b) AFF model with canards.

Table 1.

Computational grid parameters.

AFF	Coarse	Fine	With canards	Coarse	Fine
Axial	210	340	Axial	275	400
Spanwise	36	56	Spanwise	36	56
Circumferential	120	196	Circumferential	120	196
Total (Mil.)	3.96	11.27	Total (Mil.)	5.22	12.56

AFF: Air Force Modified Basic Finner Missile.

The coordinate system is shown in Figure 2. The canards and tails present a “−×” configuration initially. XY plane is the angle of attack plane. The missile spins about its longitudinal axis, which coincides with the x axis. Spin is realized through the grid motion in the inner area, and the bold line in Figure 2(a) is the interface. The positive y-axis direction is zero circumferential position. Spin rate and circumferential angle are positive when the projectile is experiencing a counterclockwise spin from the view of the base. Due to the canards interference, the aerodynamic characteristics of the tails with phase difference of $\pi$ are similar. In Figure 2(b), from the view of the projectile base, the tail at $\theta =45°$ is defined as “Fin 1” and that at $\theta =135°$ is defined as “Fin 2.”

Validation of numerical method

The inflow conditions for grid independence study were Ma = 2.5, $\alpha =12.6°$ , and $\overline{\omega }=0.025$ . Table 2 shows the relative difference between the time-averaged aerodynamic coefficients using the coarse and fine grids. The results of the two grids are in good agreement. The relative differences of the normal force and the pitching moment are within 3.5% and those of the lateral force and the yawing moment are within 9%. For subsonic calculations, grids with larger domain and amount were adopted, and the grid independence study was also conducted. Therefore, coarse grid was chosen for further calculation to obtain reliable results.

Table 2.

Relative difference of aerodynamic coefficients of different grids.

Difference (%)	Cn	Cz	Cmz	Cmy
AFF	−0.10	7.25	−0.29	0.28
With canards	1.23	−6.83	3.31	8.25

AFF: Air Force Modified Basic Finner Missile.

Figure 3 presents the time-averaged aerodynamic coefficients of the AFF configuration at different Mach numbers $(\overline{\omega }=0.025)$ . The computed results were compared with the experimental results obtained by Arnold Engineering and Development Center (AEDC). ¹³ The minimum angle of attack in the experiment was $\alpha =12.6°$ . Meanwhile, the results from different turbulence models were compared. The computed results are close to the experimental values in supersonic conditions, especially for the case at Ma = 2.5 using the $\gamma -R{e}_{\theta t}$ transition model. While, the direction of lateral force predicted by the $k-\omega$ turbulence model is opposite to the experimental situation. More results at Ma = 2.5 under different angles of attack can be found in the authors’ previous study. ⁴ In subsonic conditions, the absolute value of the computed results is smaller than that of experimental data, and the results from different turbulence models are similar. The previous study showed that the situation can be improved using the hybrid Reynolds-averaged Navier–Stokes/large eddy simulation (RANS/LES) method, while this was for the configuration with boat tail and the errors were still unneglectable. ²⁰ Although the computed results are not quite satisfied, the corresponding flow structure can be referred to for flow mechanism analysis. In conclusion, the coarse grid, the $\gamma -R{e}_{\theta t}$ transition model, and the time step $\Delta t=1\times {10}^{-5}\mathrm{s}$ were used to calculate the aerodynamic characteristics of spinning finned projectiles with and without canards.

Figure 3.

Variation of the time-averaged aerodynamic coefficients with Mach number: (a) lateral force and (b) yawing moment.

Results and discussion

Effect of canards on the time-averaged aerodynamic characteristics

The time-averaged aerodynamic coefficients of the two configurations are shown in Figure 4. Figure 4(a) presents the total normal force coefficient. The normal force coefficient increases first and then decreases as Mach number increases, reaching the extreme value in transonic condition. The total normal force of the canard configuration is larger due to the canards contribution. Figure 4(b) gives the total lateral force coefficient. For configuration without canards, the direction of the lateral force alters from negative to positive as Mach number increases, and the absolute value reaches the maximum in transonic condition. The lateral force of the canard configuration is always pointing to the negative z direction. In supersonic condition, the absolute value of the lateral force at $\alpha =12.6°$ is smaller than that at $\alpha =8°$ , because the Fin 1 and Fin 2 contribute more on the positive lateral force at Ma = 1.5 and Ma = 2.5, respectively (Figure 4(e)). Figure 4(c) shows the body lateral force. The body lateral force of the two configurations is always negative, and the absolute value increases with angle of attack. When canards are added, the magnitude of body lateral force doubles due to the asymmetric interference of canards on the flow over aftbody.

Figure 4.

Time-averaged aerodynamic coefficients: (a) total normal force, (b) total lateral force, (c) body lateral force, (d) Fin 1 lateral force, (e) Fin 1 and Fin 2 lateral force, and (f) canard lateral force.

Figure 4(d) shows the Fin 1 lateral force. The Fin 1 lateral force is always positive and reaches the maximum magnitude in transonic condition. For the configuration without canard, the Fin 1 lateral force increases with angle of attack, and the variation tendency with Mach number is regular. For the configuration with canard, the Fin 1 lateral force increases, while it presents irregular variation with angle of attack, which is possibly affected by the wash flow induced by canards. In supersonic conditions, the amplification of lateral force decreases obviously with the increase in the Mach number, especially for the case at Ma = 2.5 and $\alpha =12.6°$ . Figure 4(e) compares the lateral force of Fin 1 and Fin 2. The magnitude of the Fin 1 lateral force is larger than that of the Fin 2, while the difference of the Fin 2 lateral force at different angles of attack is more obvious. In subsonic condition, the Fin 2 and the Fin 1 lateral forces are in opposite direction at $\alpha =4°$ and $\alpha =8°$ . With the increase in the Mach number, the difference becomes smaller. Figure 4(f) shows the canard lateral force. In subsonic condition, the canard lateral force changes from positive to negative as angle of attack increases. In supersonic condition, the canard lateral force is always positive and increases with angle of attack. The canard lateral force is relatively small compared with the tail lateral force.

Figure 5 gives the time-averaged distributed lateral force in the x-axis direction for the configurations with and without canards. The area between the curves and the x axis indicates the total lateral force coefficient. The lateral force around the canards and the tails is positive, while it is negative elsewhere. In the position between the canard trailing edge and the tail leading edge, the lateral force is negative and its absolute value increases obviously due to the interference of the canard on the surface flow. For the section between the tail leading edge and trailing edge: in subsonic condition, the lateral force changes from positive to negative; the canard interference increases the lateral force at $\alpha =4.0°$ (Figure 5(a)), while decreases the lateral force at $\alpha =12.6°$ (Figure 5(b)); in supersonic condition, the lateral force is always positive, which increases first and then decreases; the variation of lateral force with angle of attack is similar to the case in subsonic condition (Figure 5(c) and (d)). In general, the body lateral force becomes dominant when the canards are added. The negative body lateral force increases obviously, while the positive fin lateral force changes just a little bit.

Figure 5.

Time-averaged distributed lateral force: (a) Ma = 0.6, $\alpha =4.0°$ ; (b) Ma = 0.6, $\alpha =12.6°$ ; (c) Ma = 2.5, $\alpha =4.0°$ ; and (d) Ma = 2.5, $\alpha =12.6°$ .

Effect of canards on the transient aerodynamic characteristics

Figure 2(b) is the state of $\phi =0°$ where the transient aerodynamic curve begins. Figure 6 shows the variation of the transient aerodynamic coefficients with rolling angles at Ma = 2.5, including the total normal force and lateral force of the configurations with and without canards. The canards interference changes the variation period of the curves from four to two within a rotation cycle, which is the same as the number of canards. Meanwhile, at the same angle of attack, the peak value of the curves increases when canards are added. The variation tendency of curves at other Mach numbers is similar, while the magnitude and amplitude are different.

Figure 6.

Variation of total aerodynamic coefficients with rolling angle at Ma = 2.5: (a) normal force, without canards; (b) normal force, canards; (c) lateral force, without canards; and (d) lateral force, canards.

The body and Fin 1 lateral force of the two configurations are shown in Figures 7 and 8, respectively. As aforementioned, the Fin 1 lateral force is dominant compared with that of the Fin 2, thus only the results of the Fin 1 are given and analyzed in detail. In Figure 7, for the configuration with canards, the body lateral force presents a sharp oscillation and alters direction with rolling angle due to the canards interference. However, the body lateral force is always negative for the configuration without canards. It can be seen from Figure 8 that the canards interference on Fin 1 is relatively weak compared with that on the body. The canards interference is stronger in subsonic conditions than in supersonic conditions. Note that the curve of the body lateral force in subsonic condition has a phase lag of $\pi /4$ compared with that in supersonic condition. The phase lag for the curve of Fin 1 lateral force is not that obvious. Next, the mechanism of the canards interference on body and tail is analyzed through flow field structure, and the essential reasons for the variation of aerodynamic coefficients with angle of attack and Mach number are revealed.

Figure 7.

Variation of body lateral force coefficients with rolling angle: (a) Ma = 0.6, without canards; (b) Ma = 0.6, canards; (c) Ma = 2.5, without canards; and (d) Ma = 2.5, canards.

Figure 8.

Variation of Fin 1 lateral force coefficients with rolling angle at Ma = 2.5: (a) without canards and (b) canards.

Interference mechanism of canards on body lateral force

As mentioned previously, the time-averaged lateral force of symmetrical configuration is totally attributed to the Magnus effect. At rolling angles $\phi$ and $180°-\phi$ , the configurations are symmetrical about angle of attack plane, and the lateral forces produced by the configuration are counteracted. However, the addition of the laterals forces at $\phi$ and $180°-\phi$ is not zero (see Figure 7), and the reason lies in the asymmetrical distortion of the flow field induced by spin.

The situations around the extreme value of the body lateral force coefficient curve (Figure 7) were chosen to analyze the interference of canards on body transient lateral force through flow field structures. Figure 9 shows the surface pressure coefficient contour and the cross-sectional vorticity magnitude of the configurations without and with canards, and the distributed body lateral force at Ma = 0.6, $\alpha =12.6°$ , and $\phi =65°$ . Compared with the case in Figure 9(a), the body lateral force in Figure 9(b) is in the opposite direction and reaches the maximum value by canards interference. The canard tip vortex is far away from the body, and its influence on the body lateral force can be neglected. Flow over the canard root affects the body pressure distribution continuously, and the high-pressure region and separation vortices shift toward the negative z direction. It can be seen from the curve of the distributed body lateral force that canards contribute more on the lateral force of the nearby body. The contribution of the canards on positive body lateral force increases gradually before the tails, while it shows no obvious change on the force of body around the tails, which can even be neglected.

Figure 9.

Surface pressure contour and cross-sectional (x/L = 0.33, 0.48, 0.72, and 0.98) vorticity magnitude of configurations (a) without canards; (b) with canards; and (c) distributed body lateral force at Ma = 0.6, $\alpha =12.6°$ , and $\phi =65°$ .

Figure 10 presents the pressure contour and streamlines of the configurations without and with canards, and the pressure difference between the right and the left bodies at x/L = 0.33, 0.48, and 0.72. The positions correspond to the cross sections in Figure 9. In Figure 10(a), for the configuration without canards, the spin rate of left-side body is against the cross-flow velocity. The cross-flow is resisted by the surface viscosity and adverse pressure gradient. Thus, the cross-flow velocity decreases and the pressure increases compared with the right side, leading to a pressure difference in the negative z direction. With the development of the cross-flow along the positive y direction, flow separation occurs earlier on the left-side body than on the right-side body, because the cross-flow cannot overcome the viscous force and the adverse pressure gradient. The right-side separation vortex stays near the surface and the cross-flow reattaches. For the cross section next to the canards, the surface pressure distribution is strong influenced by the canards. The flow separation and leeside separation vortex shift toward the negative z direction, and the low-pressure region on the right-side body decreases, leading to a positive z pressure difference; the windward surface is influenced by the flow over the canard and the high pressure shifts toward the positive z direction, resulting a negative z pressure difference around $\theta =140°$ . The case in Figure 10(b) is similar to that in Figure 10(a). The interference of the canards shifts the high-pressure regions on the leeward and windward bodies toward the right side and the left side, respectively. The pressure changes around $\theta =65°$ is also related to flow separation. In Figure 10(c), for the configuration with canards, the right-side high-pressure region and the left-side low pressure are large, leading to a positive z pressure difference, which is quite the opposite to the case for configuration without canards. For the above three cross-sections, a small positive pressure difference is produced by the separation vortices around $\theta =30°$ , while the contribution of asymmetrical separation is more obvious.

Figure 10.

Pressure contour and streamlines at different cross-sectional profiles, and the z direction pressure difference between the right- and the left-side bodies at Ma = 0.6, $\alpha =12.6°$ , and $\phi =65°$ : (a) x/L = 0.33, (b) x/L = 0.48, and (c) x/L = 0.72.

Figure 11 shows the surface pressure coefficient contour and the cross-sectional vorticity magnitude of the configurations without and with canards, and the distributed body lateral force at Ma = 2.5, $\alpha =12.6°$ , and $\phi =40°$ . Similar to the situation in subsonic condition, the body lateral force reaches the maximum value by the canards interference and alters direction compared with the case without canards. The canard tip vortex leaves away from the body quickly, while flow over the canard root obviously influences the body leeward pressure distribution. It can be seen from the curve of the distributed body lateral force that canards make continuous positive contribution on the body lateral force, which cannot be neglected around the tails.

Figure 11.

Surface pressure contour and cross-sectional (x/L = 0.33, 0.48, 0.72, and 0.98) vorticity magnitude of configurations (a) without canards; (b) with canards; and (c) distributed body lateral force at Ma = 2.5, $\alpha =12.6°$ , and $\phi =40°$ .

Figure 12 presents the pressure contour and streamlines of the configurations without and with canards, and the pressure difference between the right and the left bodies at x/L = 0.33, 0.48, and 0.72. The different positions correspond to the cross sections in Figure 11. In Figure 12(a), for configuration with canards, the left-side canard leeward expansion wave increases the body leeward low-pressure region, leading to a positive z pressure difference; around $\theta =145°$ , the left-side canard windward shock wave contributes to a negative z pressure difference. In Figure 12(b), for the configuration without canards, the asymmetrical flow separation about $\theta =90°$ makes an obvious contribution on negative z pressure difference, while the leeward separation vortices contribute slightly to positive z pressure difference. For the configuration with canards, around $\theta =30°$ , the left-side vortex is stronger than the right-side one, resulting a positive z pressure difference; near $\theta =40°$ , the right-side vortex is stronger and leads to a low pressure environment; when $\theta \approx 70°$ , flow separation occurs earlier on the left-side body and the surface pressure increases, leading to a negative z pressure difference; the asymmetrical shock and expansion waves produced by the canards are responsible for the positive and negative z pressure differences after $\theta >80°$ . In Figure 12(c), different from the vortex system in Figure 12(b), the secondary vortices are induced near the body. In the body leeward, the left-side wake vortex gets closer to the body, bringing about a low-pressure environment and a positive z pressure difference; the extreme values about $\theta =50°$ and $\theta =60°$ are caused by the asymmetrical secondary vortices. With the development of flow along the x direction, asymmetrical flow separation occurs earlier and contributes more to the negative z lateral force.

Figure 12.

Pressure contour and streamlines at different cross-sectional profiles, and the z direction pressure difference between the right and the left-side bodies at Ma = 2.5, $\alpha =12.6°$ , and $\phi =40°$ : (a) x/L = 0.33, (b) x/L = 0.48, and (c) x/L = 0.72.

Interference mechanism of canards on tail lateral force

The variations of Fin 1 lateral force with rolling angle for the configurations with and without canards are similar. To investigate the difference between the curves, subtracting the AFF Fin 1 lateral force coefficient (Figure 8(a)) from the Fin 1 lateral force coefficient of the canard configuration (Figure 8(b)) and then dividing by the maximum value of curve at the corresponding angle of attack in Figure 8(b), we obtain the contribution percentage of the canards on the Fin 1 lateral force coefficient at Ma = 0.6, as shown in Figure 13. Canards interference contributes to the positive time-averaged lateral force of Fin 1. Meanwhile, canards interference is strong when $\phi \in (0°,45°)$ and $\phi \in (225°,360°)$ . Taking the case when $\phi \in (0°,45°)$ , for example, canards interference decreases as angle of attack increases. The maximum contribution percentage decreases from 50% at $\alpha =4°$ to 10% at $\alpha =12.6°$ .

Figure 13.

Contribution of canards interference on the Fin 1 lateral force with rolling angle at Ma = 0.6.

The case for $\alpha =4°$ and $\phi =0°$ in Figure 13 is chosen for flow mechanism analysis. Figure 14 presents the Fin 1 pressure coefficient distribution at midspan, and Figure 15 shows the spatial streamlines for the configurations without and with canards. The pressure difference between the two sides of Fin 1 is smaller for the canards configuration, which means that the lateral force in the negative z direction is smaller. In other words, the Fin 1 effective angle of attack caused by the combination of cross-flow and spin rate decreases. The downwash effect induced by the canards on Fin 1 increases, which can be seen from Figure 15 for the increase in the streamline curvature.

Figure 14.

Pressure coefficient distribution at midspan at Ma = 0.6, $\alpha =4°$ , and $\phi =0°$ .

Figure 15.

Surface pressure contour and spatial streamlines around Fin 1 at Ma = 0.6, $\alpha =4°$ , and $\phi =0°$ for configurations (a) without and (b) with canards.

Figure 16 shows the contribution percentage of the canards interference on the Fin 1 lateral force at Ma = 2.5. The contribution of the canards interference on the time-averaged lateral force of Fin 1 is positive at $\alpha =4°$ and $\alpha =8°$ , while it becomes negative at $\alpha =12.6°$ . The maximum contribution percentage approaches 20% at $\alpha =4°$ , while it is only within 5% at $\alpha =8°$ . The contribution percentage reaches −15% at $\alpha =12.6°$ , namely, the canards interference contributes more to the negative lateral force, typically when $\phi \in (270°,360°)$ . In short, the canards interference is weaker in supersonic conditions than in subsonic conditions.

Figure 16.

Contribution of canards interference on the Fin 1 lateral force with rolling angle at Ma = 2.5.

The case for $\phi =315°$ in Figure 16 is chosen for flow mechanism analysis. Figure 17 presents the Fin 1 pressure coefficient distribution at midspan at $\alpha =4°$ and $\alpha =12.6°$ . The pressure difference between the two sides of Fin 1 is slightly smaller for the canards configuration at $\alpha =4°$ , and the lateral force in the negative z direction is smaller. However, the case is quite opposite when $\alpha =12.6°$ . The effective angle of attack of the Fin 1 increases, which induces a larger negative lateral force component.

Figure 17.

Pressure coefficient distribution at midspan at Ma = 2.5: (a) $\alpha =4°$ and (b) $\alpha =12.6°$ .

Figure 18 presents the Fin 1 surface pressure contour and the spatial streamlines for the configurations without and with canards at $\alpha =4°$ and $\alpha =12.6°$ , respectively. In Figure 18(a), for the configuration without canards, flow over the left-side separation vortex has a relatively strong wash effect on the Fin 1 in the positive z direction. In Figure 18(b), the spatial streamlines leave a little bit far away from the Fin 1 due to the canard interference, decreasing the wash flow effect on the Fin 1. However, when $\alpha =12.6°$ , due to the canard interference, flow over the canard root involves in the left-side separation vortex and the energy decreases. Thus, the z-direction wash flow on the Fin 1 in decreases, inducing a larger negative lateral force.

Figure 18.

Surface pressure contour and spatial streamlines around Fin 1 at Ma = 2.5 and $\phi =315°$ : (a) $\alpha =4°$ , without canards; (b) $\alpha =4°$ , canards; (c) $\alpha =12.6°$ , without canards; and (d) $\alpha =12.6°$ , canards.

Conclusion

Based on the RANS equations, combining the dual-time stepping method and the $\gamma -R{e}_{\theta t}$ turbulence model, the Magnus effect of spinning finned missiles with and without canards was investigated through numerical simulation. The canards configuration was obtained by adding two undeflected canards on the front cylindrical body of AFF missile. The time-averaged the transient lateral force of the whole missile, and each component were obtained at Ma = 0.6, 0.9, 1.5, and 2.5; $\alpha =4°,8°,\mathrm{and}12.6°$ ; and $\overline{\omega }=0.025$ . In general, the Magnus effect becomes more obvious and may alter direction for configuration with canards. The key point lies in the analysis of the canards interference on the transient and the time-averaged lateral force of body and tail through flow field structures. Therefore, the reason for the variation of the missile total aerodynamic characteristics can be explained. Conclusions are drawn as follows.

In subsonic conditions, flow over the canard root interferes with the body overflow, increasing the body boundary layer distortion at small angle of attack and the separation asymmetry at large angle of attack. The transient body lateral force alters direction with rolling angle while the time-averaged negative body lateral force doubles and becomes dominant. The canards interference has little effect on the force acting on body around the tails compared with the forebody case. The canards interference increases in the streamline curvature and the downwash effect on leeward tail increases, decreasing the negative lateral force of the tails. Compared with the configuration without canards, the time-averaged tail lateral force increases and the variation percentage of tail transient lateral force reaches 50% at $\alpha =4°$ , which both decrease with angle of attack.

In supersonic conditions, the shock and expansion waves on the canard windward and leeward interfere with the body overflow, increasing the distortion of the flow field about angle of attack plane; the leeward separation vortices and the secondary vortices contribute to an additional small body lateral force locally; asymmetrical flow separation always contributes to a negative body lateral force with considerable magnitude. The canards interference on the body around the tails cannot be neglected and the time-averaged body lateral force is smaller than that in subsonic conditions. Due to the change in the direction of wash flow as angle of attack increases, the canards contribution on the tail lateral force alters from positive to negative. At large angle of attack, flow over the canards root interferes with the leeward separation vortex and the wash flow energy decreases. The absolute variation percentage of the tail transient lateral force stays within 20% and is smaller than that in subsonic conditions.

In short, this research is concentrated on the flow mechanism of the lateral force of an axisymmetric spinning projectile with fixed canards. For asymmetric configurations with deflected canards and canted tails, and projectile experiencing complicated pitching and coning motions, the corresponding flow mechanisms need further studies.